CS 573: Algorithms, Fall 2009 Final — 7-10 PM, SC 1109, Tuesday, December 15, 2009 1. Asymmetric TSP. [20 Points] You are given the complete directed graph G = ( V,E ) over n vertices. There is an associated weight function w : E → IR + on the edges, that complies with the directed version of the triangle inequality. That is w ± x → y ² + w ± y → z ² ≥ w ± x → z ² , for any x,y,z ∈ V . Notice, however, it is quite possible that for some x,y ∈ V we have that w ± x → y ² 6 = w ± y → x ² (i.e., the weight function is asymmetric). Providing an approximation to the TSP in this case is quite harder than the undirected case, and we will tackle it in this question. (A) [5 Points] Show how to compute in polynomial time a set of vertex disjoint cycles of total minimum cost that covers all the vertices in the graph. (This is easy, but you want to be careful here.) Note, that every cycle would have at least two edges. (B) [5 Points] Let X ⊆ V be a subset of the vertices of G . Prove that there exists a cycle that visits only the vertices of X of cost at most Cost opt , where Cost opt is the cost of the optimal TSP visiting all the vertices of G . (C) [5 Points] Consider the algorithm that computes, using (A) a “cheap” cover by cycles of the vertices of the graph, it then selects a vertex from each cycle, and let Z be this set of vertices. Next, the algorithm recursively compute a cheap TSP for the vertices of Z , and it somehow generates a TSP tour for the whole graph. Describe precisely how the algorithm computes the TSP. (D) [5 Points] What is the bound on the quality of approximation provided by the TSP computed by the above algorithm? Prove your answer. 2.

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